Suppose $\frac{x}{y} = \frac 47$ and $\frac{y}{z} = \frac{14}{3}$. What is the value of $\frac{x+y}{z}$? Express your answer as a common fraction.
Answer: Rewrite $\frac{x+y}{z}$ as $\frac{x}{z}+\frac{y}{z}$. The value of the second fraction is given, and the value of the first fraction can be found by multiplying the given equations: $\frac{x}{z}=\left(\frac{x}{y}\right)\left(\frac{y}{z}\right)=\left(\frac{4}{7}\right)\left(\frac{14}{3}\right)=\frac{8}{3}$. Adding $\frac{8}{3}$ and $\frac{14}{3}$ we get $\boxed{\frac{22}{3}}$.